Inferences from the Hypothesis of Dual Electric Conduction; the Thomson Effect (1920)(en)(16s)

Voi,. 6, I1920

PHYSICS: E. H. HALL

' 1-39

2 Cohen, B. and Smith, A. H., J. Biol. Chem., 39, 1919 (489).

8 Cf. Fisher, G. and Wishart, M. B., J. Biol. Chem., 13, 1912 (49). 4 A part of the expenses of the work herein reported has been defrayed from the Francis E. Loomis Research Fund of the Yale University School of Medicine. More extensive reports will be sent to the Journal of Pharmacology & Experimental Therapeutics and to the Archives of Internal Medicizze.

INFERENCES FROM THE HYPOTHESIS OF DUAL ELECTRIC CONDUCTION; THE THOMSON EFFECT By EDWIN H. HALL JZ1*tItRSON PHYSIcAL LABORATORY, HARvARD UNIVERSITY Communicated January 29, 1920

At the Washington meeting of the National Academy of Sciences in April, 1919, I presented two papers that have not yet been published. One was on the Effect of Pressure on Electric Resistance and on Peltier Heat in Metals, the other on Thermal Conduction in Metals, both being written from the standpoint of Dual Electric Conduction. The first named of these two papers contained implicitly the following propositions: la. Increase of pressure should, by bringing the atoms and the metal ions closer together, increase ka, the associated-electron conductivity, and decrease kf, the free-electron conductivity. We might, then, expect the total conductivity, k, to increase under pressure in metals having a relatively small value of (kf . ka) and to decrease in metals having a relatively large value of this ratio. lb. As antimony and bismuth have exceptionally small values of k, they probably have exceptionally large values of (kf . ka), and this may account for the fact that, among twenty metals examined by Bridgman, these two were the only ones to show a decrease of conductivity under an increase of pressure. 2a. If the ratio (kf . ka) is greater in metal B than in metal A, ionization must occur at the junction of the two metals when a current flows from A to B, and re-association must occur there when the current flows from B to A. As ionization is doubtless accompanied by absorption of heat and re-association by evolution of heat, we have here an action which may play a very important part, if not the chief part, in the Peltier effect. 2b. The exceptionally large value of (kf . k.) that probably exists -in bismuth may account for the fact that heat is absorbed when a negative current goes into this metal from any other. . 2c. As increase of pressure 'probably decreases the ratio (kf k+), flows a current where of heat negative an absorption *we should expect If the same to metal uncompressed. under pressure high metal from a we the effect and minus, of opposite effect this plus compression we call

I40

PHYSICS: E. H. HALL

PROC. N. A. S.

find that, among eighteen metals examined by Bridgman under high pressures, thirteen showed plus effects only, none showed minus effects only, but five showed mixed effects, minus at 00 C. and plus at 1000 C. 3a. According to the theory under discussion the Thomson effect should disappear if (kf . ka) became zero. Accordingly increase of pressure, causing a decrease of this ratio, should diminish the Thomson effect. Among eighteen metals examined by Bridgman under pressure, nine showed a decrease throughout the whole range of observation, one showed an increase through the whole range of observation, one showed zero change everywhere, the other seven showed mixed effects. The following may be taken as a summary of the second paper: 1. Thermal conduction in a metal may be due to the convective action of a circulating electric current, free electrons moving down the temperature gradient and associated electrons moving up, with ionization at the hot end of the metal, involving absorption of heat, and re-association at the cold end, involving the emission of heat. 2. A quantitative test of this thesis, made necessarily with various assumptions, indicated ionizing potentials of the same order of magnitude as those observed in the ionization of metal vapors; but it seemed doubtful whether values so large as those here indicated are consistent with the magnitudes of the Thomson effect. It is to be noted that ionization within the solid body of a metal may well require less energy than ionization of the vapor. It seemed desirable to study the Thomson effect more fully from the standpoint of dual electric conduction before proceeding farther with the theory of thermal conduction. The results of an examination o. the Thomson effect are now ready for publication. THE THOMSON EFFECT In what follows: n = the no. of free electrons per cu. cm. of a metal. = the no. of cu. cm. of metal containing 1 gm. of free electrons. v m = mass of electron, and G = no. of electrons in 1 gm. of electrons. Then nv = G = 1 . m. (1) p = press. of free electrons, in dynes per sq. cm. of cross-section of the metal. R = the gas constant for one molecule, or for one electron, = 1.37 X 10-16. Then p = nRT, (2) (3) and, for 1 gm. of free electrons, pv = nvRT = RT . m. P = electro-mag. pot. due to purely electric forces within the metal. Pa = electro-mag. pot. due to differential attraction of the unequally heated metal for the associated electrons. P1 = corres. pot. due to differential attraction of the metal for the free electrons.

VOL. 6, 1920

ka

kf k e X

A'

PHYSICS: E. H. HALL

141

= part of specific electric conductivity due to associated electrons. = part of specific electric conductivity due to free electrons. = ka + kf = total electric conductivity. = the electron charge in electro-mag. measure = 1.6 X 10-20. = the latent heat of ionization per (1 + e) electrons, in ergs. = the latent heat of ionization per electron, in ergs.

Hypothesis (A) is, that the mere mechanical tendency of the free electrons is toward uniformity of pressure throughout the unequally heated metal. Hypothesis (B), alternative with (A), is that the mechanical tendency of the free electrons is toward the state of equilibrium produced by thermal effusion; that is, = constant. p + (4) Let C of figure (1) be the cold end and H the hot end of a metal bar forming part of a circuit in which an electric current is maintained by thermo-electric action, the resistance of some part of the circuit being so great that the conditions existing in CH are very little different from those of'equilibrium. In this case the Joule heat generated in CH can be neglected in comparison with the Thomson heat there generated or absorbed. Contrary to custom, the direction of the stream of electrons through the metal will, in this paper, be taken as the direction of the current, and accordingly a, the Thomson heat at any temperature T, will be defined as the heat absorbed by the electromagnetic unit quantity of electricity, (1 + e) electrons, in going through the metal from a place of temperature (T - 0.5) degree to a place of (T + 0.5) degree. This definition will make a negative for copper and positive for iron. The value of of will be expressed in ergs.

clIIH1

d-

dT

Fio. I

When the unit quantity of electricity, (1 . e) electrons, (m + e) gm., goes through the slice dl of the bar CH, from the isothermal surface T to the isothermal surface T + dT, the fraction (kf +. k) of it consists of free electrons and the part (k% . k) of associated electrons. We -have now to take note of the changes of energy, of various kinds, undergone' by these two parts of the current. We shall list the various forms of energy here considered under five general heads: (1) bulk potential energy, or pv potential energy, to which the free electrons only are subject; (2) kinetic energy of the electrons, which we shall regard as negligible in the associated electrons and equal- to that of monatomic gas molecules in the free electrons; (3) electric-charge potential energy, the P energy, to which both the free and associated electrons are alike subject; (4) the Pf potential

PHYSICS: E. H. HALL

142

PROC. N. A. S.

energy, which affects free electrons only; (5) the Pa potential energy, which affects the associated electrons only. We must take account also of the gain of energy involved in the ionization which. may occur in the current from T to T + dT, due to the increase in the ratio (kf . k) with rise of temperature. Under hypothesis (A): All changes due to change of (kf + k) being considered last, we have as the change of pv energy, (see equation (3)) (kf . k)(m - e)d(pv) = (kf + k)(R . e)dT. (a) The gain of thermal kinetic energy by the free electrons is (kf . k)(1 +. e) RdT. (b) 2

The gain of P potential energy is dP. (c) The gain of Pf potential energy is (kf + k)dPf. (d) The gain Of Pa potential energy is (ka + k)dPa (e) The gain of energy through ionization is Xd(kf + k) = (1 ÷. e)X'd(kf - k). () The sum of all these quantities is the Thomson-effect heat absorbed between T and T + dT; that is adT =(a) + (b) + (c) + (d) + (e)+(f) (5) From this we get _kf 5R ____d(k (6) _dPf k~ dPa dP e k 2e LkdT k *dT dT dT From the conditions of equilibrium, under hypothesis (A), in a detached bar like CH we have' kf*e(R+ RT dn +[kf*dPf+ka dPaT d] =0 (7) k- ekt&dT,LdT k dT dTJ Subtracting (7) from (6) we get (8) kf 3 R kf RT dn + V d(kf-k) 8 k2 eknedT dT e Under hypothesis (B) we get 0.5R instead of the first R in (7), and this gives 2R instead of 1.5R in the first term of (8). Hypothesis (B) makes no other change in (7) or (8). I shall now try to put equation (8) into a form suitable for dealing with the values of a found by Bridgman2 in his experiments on a large number of metals. In this undertaking I assume that for present purposes the following equations hold above 00 centigrade: n = zTq, (9) and (kf * k) = C + Clt + C2t2, (10) and are and t is where z, q, C, C, C2 constants, temperature on the ordinary centigrade scale.

VOL. 6, I1920

PHYSICS: E. H. HALL

143

I shall, moreover, assuming that the total heat of ionization per electron is made up of a part X'0, due to the overcoming of atomic attraction, a part 1.5RT for the kinetic energy gained, and a part RT for the pv potential energy gained, write (11) '= '0 + 2.5RT = X'0 + 2.5R(273 + t). Keeping to hypothesis (A), and so using equation (8) for a-, I get by substitution according to eqs. (9), (10) and (11), (12) a- = K + (Ki + K20 T, where K, K, and K2 are constants,3 defined by the equations

K =

-[C(l.5-q) + CLQ R-273(1.5-q)) - 273C2(2j1- 273(1.5

K1 =

-[Cj(4 - q) +

.C2(2

-

q))] (13)

-273(1.5 - q))]

(14)

R (15) K2 = -e C2(6.5 - q). I have put a- into the form shown by equation (12) in order to make my expression for it correspond as nearly as may be to that used by Bridgman to set forth the results of his experiments. He writes, in substance, a = (A + Bt)T, (16) where A and B are constants, the latter being zero in many metals. Bridgman finds nothing corresponding to my constant K, and I have spent much labor in attempting to get rid of this constant; but no reasonable assumption that I can make eliminates it from my general expression for a. On the other hand, equation (13) shows that K is the sum of many terms, some positive, some negative, and there is nothing to show that it may notebe very small, too small to appear in such experiments as those of Bridgman. Accordingly, in dealing with his observations I put K, and so the second member of (13), equal to zero. This gives me an equation of which I make frequent use in the form

C= [-Ci(-273(1.5-q)) + 273C2(

R273(1-5-q))]

(1.5-q). (17) As to the A and B of equation (16), I take these to be, respectively, the K1 and the K2 of my equations, and, as Bridgman gives the value of A and B for every case dealt with, I have the K, K1 and K2, of equations (13), (14) and (15), replaced by definite numerical terms. These three equations now contain the five unknowns, C, C1, C2, Xfo, and q. Accordingly, I must assume values for two of these quantities in order to evaluate the other three. As a rule, I have assumed values of

PHYSICS: E. H. HALL

I44

PROC. N. A. S.

q and X'0, different ones in succession for each metal, and have then worked out the corresponding values of the other quantities, following the order C2, Cl, C. Values of q and X'0 that would lead to values for C greater than 1 or less than 0 are of course rejected, as such values of C would be meaningless, but negative values of Ci and C2 are not to be regarded as impossible. I was at first inclined to the opinion that the ratio (kf *+ k) would always increase with rise of temperature, but this is not a logical necessity, in the present state of our knowledge, and it appears from what follows that the ratio in question is quite as likely to decrease as to increase in the temperature ascent from 00 to 1000. An interesting relation between this conclusion and the observations of Bridgman on change of resistance under pressure, at various temperatures, will be shown in this paper.

Cases in which K2, or B, is 0: In twelve of the seventeen metals for which Bridgman gives the value of a the B of equation (16) is zero. Such cases are very easy to deal with. We have K2 = 0, and so, from equation (15), C2 = 0, unless q has the improbably large value 6.5. If C2 is 0, we have, from equation (14),

C,

=

K, (or A) R (4 -q).

(18)

Substituting for Ci in equation (17), we get

C= [-K .R(R-273(1.5-q)) (4-q)] (1.5-q). (19) For any given value of q this becomes C = K'X'o- K" (20) where K' and K" are new constants, the values of which depend on q. This equation shows that, for a fixed value of q, we can represent the

x

PIG. 2

relation of C to X'o by means of a straight line drawn on the (C-X'o) plane. Such a line is useful for purposes of interpolation and extrapolation. It is to be noted that the metals for which K2 is 0 fall into two groups, for one of which K, is positive, while for the other it is negative. Figure 2 shows the general character of the set of q-constant lines for the first group, and figure 3 does the same for the second group.

Voi,. 6, I1920

PHYSICS: E. H. HALL

145

For both of these groups 1.5 is a critical value for q. Examination of equation (19) shows that, when q = 1.5, C becomes infinite unless XI. at the same time becomes 0; and if X'0 becomes 0 while q'= 1.5, C becomes

FIG. 3

indeterminate. In both figure 2 and figure 3, therefore, q = 1.5 would imply a line coincident with the C axis. If q in equation (19) has a value between 1.5 and 4, C will have the same sign as K1, which is positive for the first group and negative for the second group. Accordingly, since negative values of C are meaningless, 1.5 0 and K2 = 0 TABLE 1:

COBALT.

K1 = 7.8, K2 = 0, C2 = 0 Ci=228X10'- Ci=261X1O6 C,

Ci = 305 X 10-*

Ci = 457 X 10-6

Ci = 915 X 10-6

or

(kf k), kaytko)

(kf *1 k) at

(kf . k)

at 100

q

(kf

at

k)

100'

(kf

a 0

at

k)

100'

a q

o

(kf

k)

at 1000

3 0.01 0 .03 .033 .5 .02 .036 1 .01 .041 2 1 .00 .111 2 3 .5 0.081 0 2 .01 .246 3 1 .5 0.20 0 2 .02 .346 3 .01 .392 1 .50.30 0. 2 .03 .446 3 .02 .492 1 _ 0.40 0 .5 1 If there is any combination of q and 6, that will make C = 0.10 for cobalt, the q

must be very near 1.5.

PHYSICS: E. H. HALL

VoL. 6, I920

'47

TABLE 2: NICKEL.

K1

3.56, K2

=

0

=

Ci- 139 X 10-'

Ci- 119 X 10-6

Ci- 104 X 10-S

0, C2

=

Ci

-

Ci - 416 X 10-'

208 X 10-'

C, or

(kf k), -

at 00

*o

a

.o

_

(kf+k))

(kf+L-k)

°

a

(kf+-Ok)

*,

at 1000 q

80

q

(kf+-lk)

*~atl00

3 0 .02 .020 .5 .016 .022 1 .01 .024 2 1 2 .010 .121 3 0 .5 1 2 .033 .221 3 .03 .242 0 .5 12 .056 .321 3 .06 .342 .5 0 __ 2 .079 .421 3 .09 .442 1 0 .5

0.01 0.10 0.20 0.30 0.40

-

PALLADIUM

K1 = 3.52, K2 = 0, C2 = 0 Palladium is so like nickel in its Thomson effect that table 2 will serve for it.

TABL], 3: PLATINUM. 2.67,- K2 = 0, C2 = 0

K1 C, or

(kf ÷k), at 00

Ci- 78X 10-6 q

|

(kf

0

-

k)

Ci q

-

=

Ci

89 X 10-6

|0

(kf+ -tk)

at 1000

104 X 10-' Ci

=

(kf1000 k) * at

0

|

=

156 X 10-6

q-1a o(kf+k) at100-

0.01 0 .02 .018 .5 .01 .019 1 .01 0.20 21 2 .02 .116 .5 0.10 0o1 . 52 .05 .216 0.20 0 12 .08 .316 .50.30 0 1.5 2 .11 .416 0.40 0 -

TABLE 4:

K1 = 0.134, K2 C1= 392 X 10-8 Ci 448 X 10-' C,

or

at(k+k) 0

0.01 0.10 0.20 0.30 0.40

_

_

_

_

___

_

_

_

-

-

_

1 1 1 1

|o

atf 100°

33 .01 .131 3 .05 .231 3 1.09 .331

3 .13 .431

-

-

Ci 780 X 10-' Ci

1560 X 10-8

_

at 100

-1 -

312 X 10-'

0, C2 = 0

(kf'÷k)

.

at 11000

.5 .5 .5 .5 .5

-

TIN. =

Ci 520X 10-8

_

(kf +k)

(kf +k)

at 1000

0 0 0 0 0-_

_

Ci

. a

0.05 0.52 1.04 1.56 22.08 2 2 2 2

(kf*÷k) at

0

1000

.011 .101 .201 .301 .401

3 3 3 3 3

at 100

0.05 .012 0.79 .102 1.62 .202 2.45 .302 3.28 .402

148

PROC. N. A. S.

PHYSICS: E. H. HALL TABLE 5: ZINC.

K1 = 0.99, K2 = 0, C2 = 0 (kf1k),

-

Ci - 116 X 10-*

Ci - 58 X 10-

38 X 10-6

_

(k +k) at 1000

at 00

0.01 0.10 0.20 0.30 0.40

Ci

33 X 10

Ci

Ci 29X 10-

s

0 0000-

.5 .5 .5 .5 .5

(k1+k)

(kf +k)

0

-

-

-

*

o

at 1000 Q

1 .0 1 1 11-

.014 2 2 2 2 2

at 1000 q

0

0

(kf+k)

(kf+k)

*

at 1000

at 1000

0

3.106 3 .07 .206 3 .18 .306 3 .30 .406 3 .41

-

.06 .13 .20 .28

.112 .212 .312 .412

Second Group: Metals for which K1 0 and K2> 0 TABLE 12: IRON.

K1 = 1.78, K2 = 0.0516 C2 93 X C,

C2 = 110X 10-8

100Xl0-8

C

or_

(kf +-Lk),

8~C1 x 10*1 at(kfp-.'k) 100

at 10O0

80 AO

0

0 0 0 0 0 0 0.25 0

0.001 0.01. 0.05 0.10 0.15 0.20

.017 .024 .057 .101 .146 .192 .237

66 44 -21 -81 -135 -178 -221

.015 .019 .031 .042 .052 .060 .068

.5 .5 .5 .5 .5 .5 .5

C

.010 .012 .019 .027 .034 .039 .045

X 10_6

(kf+li')

70 57 10 -43 -90 -123 -163

.018 .026 .061 .106 .151 .198 .244

c1 X 10-* (kf+k)

*so

at 100

1 1 1 1 1 1 1

at 1000,

.005 .006 .009 .013 .017 .020 .022

.020 .028 .065 .112 .158 .205 .254

77 68 43 9

-26 -52 -69

TABLE 13: THALLIUN.

K1 = 0.268, K2 = 0.00336 C2 = 6 X 10-8 C,

(kf

C2-= 7.1 X 10-8

C2= 6.5 X10-8

or

k),

at0

0.001 0.01 0. 05 0.10 0.15 0.20 0.30

Ci X10-*1

ato100

7 -3 -32 -51 -65 -78 -99

.002 .011 .047 .096 .144 .193 .291

0 -0 0 0

.020 .048 .130 .186 0 .227 0 .262 0 .324

'1'."'

a

at 1000

.5 .010 .5 .034 .5 .090 .5 .130 .5 .160 .5 .186 .5 .220

so410000~0at

E* 1000

10 -1 -25 -42 -55 -66 -81

1000

.003 .011 .049 .098 .147 .196 .295

10 .003 1 .006 .011 1 .016 5 .048 1 .050 -14 .096 1 .080 -31 .145 1 .096 -40 .194 1 .105 -45' .293 1 .135 -62

Fourth Group: Metals for which K1 0 TABLE 14: ALUMINUM.

K1 = -0.016, K2 = 0.006 C2

=

11 X 10-8

C2

=

12 X 10-8

C2

=

13 X 10-8

Ci

X

C, or

(kf 00k), L

at

q

Bo

I Ci X 10-6

-1 0.001 0 .020 0.01 0 .036 -11 0.05 0 .086 -41 0.10 0 .125 -64 -97 0.20 0 .180 0.30 0 .226i -125

(kf at 1000 q -k)

. 002 .

5o

.5 .012 .010 .5 .022 .047 .5 .058 .095 .51 .088 .191 .5 .126 .289 .5 .155

Ci X 10-6

(k .l k) at 1000

q

-1 -9

.002

1

.010

1

-37 -61 -91 -114

.048

1

.095

1

.192 .290

1

1

108'

-1 .006 7 .012 .032 -30 .047 -46 .068 -69 .088 -91

(kf - k) at -1000

.002 .011 .048

.097 .194 .292

PHYSICS: E. H. HALL

VOL,. 6, 1920

ISI.

TABLE 15: GOLD.

K, =-0.934, K2 = 0.001 or

(kf k), 0 at 0

C2

C2-- 2.0 X 10-

-CT- 1.8-X 10C,

-

2.1 X 10-'

--~

O°

s

0.001 0 .032 0.01 0 .051 0.05 0 .122 0.10 0 .188 0.20 0 .292 0.25 0 .334

|

1a0-

|

(k1+.k) at 1000

-29 -31

.007 .046 .096 .195 .244

-38 -45 -56 -60

iD

B

cxo'(kf+'k) 10fw |

at 100

.5.024 .5 .040

.5.096 .5 .158 .5.256 .5 .294

-33 -35 -42 -50 -62 -77

1 .007 1 .046 1 .095 1 .194 1 .293 1

*

(kf+k)

CX-

B

at 1000

.011 .018 .050 .083 .136 .158

*-38 -39 44 -50 -58 -64

-

.006

.046 .095 .194 .244

TABLE 16: MOLYBD1ENUM.

K, = -4.334, K2 = 0.015 C2 =27 X 10-8 C,

or

atO0 |

0.001 0.01 0.05 0.10 0.20

_

_

_

_

0 C,|1(l

0 .025 -138 0 .030 -146 0 .047 -173 0 .064 -197 0 .094 -220

C2 -29 X 10_

C2

-

32 X 10-

_

(kf -k)

at 1000

|kf

ClX10'

.5 .017 -155 .5 .020 -161 .035 .5 .030 -180 .083 .5 .042 -204 .181 5.5.061 -240

ClX k

(kfk)

at 1000q8,

1 1 .035 1 .083 1 .179 1

.008 .010 .015 .0201 .030

(kf+ k)

at 100

-175 -180 -192 -205 -230

.044 .083 .180

Comments Lead does not appear in these tables, for the reason that a for this metal is so small that we have no formula for it. Zero value for o- could be accounted for, according to the principles of this paper, in either of two ways: First, as equation (8) shows, (kf + k) might be zero. Second, if in equation (8) we make (kf +. k) a constant, of whatever value,4 and is zero if n oc T.' These tables, like the experimental values of af by means of which they are made, have no great pretension to accuracy. For example, it is doubtful whether much confidence can be placed in the small value of K2, very nearly 0.001, found by Bridgman for gold. Yet the effect of neglecting this value is considerable. Thus, if we call K2 = 0 for gold, we get

PHYSICS: E. H. HALL

152

PROC. N. A. S.

TABLE 15 bis: GOLD.

-0.934, K2 taken to be 0, and C,

K,

(kf

or + k).,

_

atoat 0|ek| 0 0.01 0.10 0.20 0.40

_ _

0 0 0 0

_ _

_

0 C --36 X 10-

C =-31 X 10-

C =-27 X 10C,

=

_

_

_

_

80

(kf+k) 1OO0 | ~~~~at

0.08 0.51 0.99 1.95

0.007 0.097 0.197 0.397

_ _

_ _

0.5 0.5 0.5 0.5

_-_

_

_

0.05 0.30 0.58 1.14

_

_

_

at

(kf+--k) 10O0

q

0.007 0.097 0.197 0.397

1 1 1 1

_

_-_

_

_

_

| (kf atlOO0k) 0.02 0.13 0.25 0.49

0.006 0.096 0.196 0.396

Comparison of this with table 15 shows that ignoring the value of K2, small as it is, makes a good deal of difference in the values of 5., though comparative little in the values of (kf . k) at 1000, as calculated by means of equation (10). Uncertain as the values of (kf *. k) are, there is little room for doubt in most cases, if my theory is substantially sound, as to the direction of change of these values when the temperature is raised. Accordingly I have divided all the metals represented by the preceding tables into two groups, in the first of which (kf * k) is greater at 1000 than at 0°, while in the second group the ratio in question is greater at 00. For most of these metals Bridgman has determined the pressure-coefficient of resistance at 00 and 1000. For magnesium the value at 00 only was found; for bismuth the highest temperature used was 75°. For all of the metals here considered except bismuth the coefficient in question is negative-that is, the resistance decreases with increase of pressurebut for bismuth it is positive. I shall make use of what Bridgman calls the average pressure-coefficient, the average value of the coefficient through a range of pressure from 0 kgm. to 12000 kgm. per square centimeter. I shall let 7ro represent the value of this coefficient at 00, and 7r,oo the value at 100 0. In accordance with what has been said in the opening paragraphs of this paper we should, other things being equal, expect 7r to decrease, numeri ally, with increase of (kf . k) in metals for which -r is negative, and to increase with increase of (kf + k) in metals for which r is positive. Accordingly we might expect (irloo -7ro) + 7ro to be, in general, a negative quantity for metals in which (kf +. k) increases with rise of temperature, and a positive quantity for metals in which (kf . k) decreases with rise of temperature-bismuth of course requiring exceptional consideration. The table given below enables us to test the validity of this expectation.

PHYSICS: E. H. HALL

Voi. 6, 1920

1153

TABLE 17

(A) MITAIs N wicH (kf + k) INCRUASS WITH RISE or TuMPuRATuR re

mios

(B) METAS IN WMCH (kf + k) DZCREASES wI INCREASE OF T1xmPERATURiI

(WItoo-) wo.

Co -0-.0873 -0.0.726 Ni -0O.0,147 -O. 0.158 Pd -0.06190 -0.0.186 Pt -0.05187 -0.05184 St -0.0.920 -0.06951 Zn -0.0,470 -0.06454

-16.9% +7.5% -2.1% -1.6% +3.4% -3.4% Mean -2.2%

we

Ag Al An Bil Cd Cu Mg2 Mo Ta W

-0. 0333 -0. 0382 -0. 05287 +0. 04223

WI

(ose-w) +w

-0. 0336 -0.0.377

+0. %e -1.3%

-0.05292

-0. 0183 -0.0,177

+1.7% +9.4% +3.7% -3.3%

-0.0,55 0.05129 -0.06127

-1. 6%

-0. 04115 -0. 04123

+7.0%

+0. 04202

-0.06894 -0.0,927

-0.0,123 -0.0,126

+2.4% Mean +2.1% 1 For bismuth, because T is positive and the highest temperature for ir was 750, we use (rO - T76) .- 0. 2 Bridgman did not find irsoo for magnesium.

Iron does not appear in this table, for the reason that, as table 12 shows, it should go into Section (A) under some conditions, but into Section (B) under other conditions. Its value of (7rioo - 7ro) + 1r is about 4%. This value, if put into Section (A), would make the mean value there -1.3%; put into Section (B), it would make the mean there + 2.3%. If (kf + k) for iron is greater than 10% at 0°C., which seems likely, iron belongs to Section (B). Thallium and aluminium should go into Section (A) if (kf . k) in them is less than 1% at 0WC; but this is improbable. It is to be observed that for each section of table 17 the average value of (7rioo -7ro) + 7rO comes out with the sign it should have according to the predictions of the dual theory of electric conduction, as used in this paper, a minus sign for Section (A) and a plus sign for Section (B). This can hardly be pure accident. Under Hypothesis (B) All of the equations and all the tables of this paper, thus far, are based upon or are consistent with "hypothesis (A).'" If hypothesis (B) is adopted instead, equation (8) is changed in the manner already described and the result in equations (13) to (19) is to replace 1.5 by 2, 4 by 4.5, and 6.5 by 7, in the parenthesis (1.5 - q), (4 - q) and (6.5 - q). The resulting tables, for cobalt and bismuth, the only metals for which the calculation has been made, are given in Tables 18 and 19 (below). By comparison of these two tables with the corresponding ones obtained by the use of hypothesis (A) we -see that no radical difference in the re-

PHYSICS: E. H. HALL

54

PROC. N. A. S.

sults comes from the substitution of one hypothesis for the other. Hypothesis (B) seems to give, other things being equal, somewhat larger values of 5. than hypothesis (A). TABL3 1[8: -COBALT.-

K = 7.8, K2 = 0, C2 = 0 Ci =202 X 10-' C

(kf

C= 230 X 10-6

Ci= 303 X 10-

Ci = 260 X 10-

or k ),

at 0o0

0.01 0.05 0.10

(kf-k)

k) (kf100000 at

5O

q

.029 .007

.038 .030 0.5 .004 .070 0.5 -0.5

0 0 0

a

at

.033 .073

(kf 100k) at

1

.020 .036 1.5 .007 .076 1.5 --1.5

.040 .080

.010 .005 -

Ci= 607 X 10-6

Ci= 455 X 10-6 C

k) (kf at 1000

a

or

at 0

|

0.10 0.15 0.20 0.30 0.40

| ~~~~~~at (kfI+k) 0 I000

ql

2.5 2.5 2.5 2.5 2.5

0.196 0.246 0.346 0.446

0.002 0.007 0.016 0.026

(kf

so

k)

at 1000

-

3 3 3 3 3

0.005 0.019 0.032

0.261 0.361 0.461

TABLE 19: BISMUTH.

K1 = -3.2, K2 = 0, C2 = 0 Cl =-83 X 10C.,

Ci =-93 X 10--'

Ci=-106 X 10-6

or

(kf + k), _

-

(kf1k)

at °°

q

so

(kf +00k) at 1000

q

0.01 0.10 0.20

0 0 0

0.07 0.25 0.46

0.002 0.092 0.192

0.5 0.5 0.5

0.40

0

0.87 0.392 0.5 0.58 0.391

0.60

0

1.28

0.592

0.5

8

0

at 1000

0.05 0.001 0.091 0.191

0.17 0.32

0.84

0.591

a

a

(kf

k)

at 1000

0.03 0.000 0.10 0.089 1.5 0.18 0.189 1.5 0.34 0.389 1.5 0.50 0.589 1.5 1.5

In a paper already well advanced I shall undertake to show how far the data obtained in the present paper enable us to go in the way of explaining thermal conduction in the metals here dealt with. 1 Eq. (7) is obtained from eq. (1) of my paper in the Proceedings of the National Academy of Sciences for April, 1918, by substituting for , from eq. (2) of that paper (which equation should have n instead of m) and then making obvious changes. 2 Proceedings of the Amer. Acad. of Sciences, Vol. 53, No. 4, 1918, pp. 269-386. These values are not regarded by Bridgman as accurate, having been obtained as second derivatives of e. m. f. values, the quantities measured; but taken as a whole they seem to be the best available data for the present purpose. 9 This assumes that V' is constant. 4 When (kf ÷k) is infinite, we have the conditions discussed in my paper "Thermoelectric Diagrams on the P-V Plane," Proc. Amer. Acad., Boston, February, 1918.